$L^2$-complexes and twistor complexes of tame harmonic bundles

TL;DR

This paper proves a version of the Hard Lefschetz theorem for certain twistor D-modules on Kähler manifolds, extending Hodge theory results to a twistor setting with applications to complex geometry.

Contribution

It establishes the Hard Lefschetz theorem for push-forwards of regular polarized pure twistor D-modules, generalizing classical Hodge theory results.

Findings

Proves the Hard Lefschetz theorem for twistor D-modules on Kähler manifolds.

Develops a twistor analogue of Kashiwara and Kawai's theorem on Hodge structures.

Extends Hodge theory to a broader twistor framework.

Abstract

Let $f:X\to Y$ be a morphism of complex manifolds. Suppose that $X$ is a K\"ahler manifold. Let $(\mathcal{T},\mathcal{S})$ be a regular polarized pure twistor $\mathcal{D}$-module of weight $w$ on $X$ whose support is proper over $Y$. We prove the Hard Lefschetz Theorem for the push-forward of $(\mathcal{T},\mathcal{S})$ by $f$. As one of the key steps, we obtain the twistor version of a theorem of Kashiwara and Kawai about the Hodge structure on the intersection complex of polarized variation of Hodge structure.